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On the asymptotic behavior of the solutions of third order delay differential equations. (English) Zbl 1321.34097

The authors consider the nonlinear third order differential equation with constant delay \[ [\Psi (x){x}'{]}'' + a(t){x}'' + b(t)\Phi (x){x}' + c(t)f(x(t - r)) = e(t), \tag{1} \] where \(r \in \mathbb R , \mathbb R = ( - \infty ,\infty ),r > 0\) is the constant delay, \(t \in \mathbb R ^{+}, \mathbb R ^ + = [0,\infty ), \quad a(t),b(t),c(t),e(t),\Psi (x),\Phi (x)\) and \(_{ }f(x)\) are continuous functions, and the derivatives \({\Psi }'(x),{\Phi }'(x)\) and \(_{ }{f}'(x)\) exist and are continuous for all \(x \in\mathbb R\).
The authors give sufficient conditions for the asymptotically stability and boundedness of the solutions of equation ({1}), when \(e(t) = 0\) and \(e(t) \neq 0\), respectively. By defining a suitable Lyapunov functional, they prove two new theorems on the subject. The obtained results extend the results in the literature. They also give an example for the illustrations.
Reviewer: Cemil Tunç (Van)

MSC:

34K25 Asymptotic theory of functional-differential equations
34K20 Stability theory of functional-differential equations
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